FIG. 1 illustrates the minimum variance frontier of risky assets. FIG. 1 is based on FIG. 7.10 from the Bodie reference described below. Conventional Mean-Variance Optimization (MVO), as pioneered by Harry Markowitz, is well-known and has been used to construct portfolios for the last half-century. For an exposition see, for example, Investments, by Zvi Bodie, Alex Kane, and Alan J. Marcus (Irwin, (c) 1996), Section 7.4, “The Markowitz Portfolio Selection Model”. Also see Portfolio Optimization, by Michael J. Best, (Taylor & Francis, (c) 2010). Under MVO assumptions, higher risk is associated with higher returns, leading to the well-known “efficient frontier”
FIG. 2 illustrates the risk-return plot for two overlapping 20-year time periods. FIG. 2 is based on FIG. 4.6 of Bernstein reference described below. MVO is known to be unstable in practice: see, for example, The Intelligent Asset Allocator by William Bernstein, (McGraw Hill, (c) 2001). Some “hints” (constraints) usually must be supplied to prevent the algorithm from chasing asset sectors with recent high performance. Another practical consideration is that in many cases investors are limited to long-only positions, either because of difficulties in obtaining credit for short positions, or because short positions are forbidden by statute or by investment policy (e.g. in some retirement accounts and pension plans).
Despite the theory, results from historical data tell a different story. The so-called “Low Volatility Anomaly” (LVA) has been documented extensively. See for example The New Finance—The Case Against Efficient Markets by Robert A. Haugen, (Prentice-Hall, (c) 1995). Also see: http://www.spindices.com/documents/presentations/20121106-global-webinar-spdji-kang-look-volatility.pdf
The data show that in many cases, a lower volatility stock portfolio is associated with higher returns: this is NOT what the well-known MVO risk/return tradeoff would have suggested.
After a number of years during which the existence, magnitude, and possible causes of the LVA have been debated, some recent ETF (Exchange-Traded Fund) offerings have been developed to market it. See for example: http://seekingalpha.com/article/1230991-low-volatility-landscape-gets-4-new-etfs and http://www.invescopowershares.com/products/overview.aspx?ticker=XSLV
One suggestion for incorporating this phenomenon into practical portfolio construction (as advocated for example by Haugen, for example, in Chapter Six of his book, Section “Value Man and Growth Man Too”) is to use MVO “top-down” to determine a broad stock vs. bond mix, but to then use a low-volatility criterion for detailed stock selection.
Alternatively, risk-aversion and time horizon considerations can also be used to determine the top-down asset mix. See “Strategic Asset Allocation” (John Y. Campbell and Luis M. Viceira, Oxford University Press, (c) 2002), Chapter 3, esp. FIG. 3.3.
Small-cap stocks are especially attractive in the context of portfolio construction since they have historically enjoyed a long-term performance advantage over large-cap stocks. See for example the SBBI (Stocks, Bonds, Bills, and Inflation) 2006 Yearbook (Ibbotson Associates, (c) 2006) which shows that over the period 1926-2005, the geometric mean return on large company stocks was 10.4%, while for small company stocks it was 12.6%, giving small-cap stocks a 2.2% annualized advantage.
FIG. 3 illustrates the effect of moving to lower- and higher-risk portfolios. FIG. 3 is based on FIG. 6.16 from Haugen and shows that the estimated annual advantage of a low-volatility portfolio (what Haugen refers to as the “efficient version”) is about 1%, and so, assuming additivity, approximately 3.2% of excess annualized performance (on average) is achievable.
FIG. 4 illustrates value versus growth in the U.S. from 1927-2006. FIG. 4 is based on FIG. 8.20 of the Hebner reference described below. Depending on the data source, it may in fact be that the improvement in returns is more than additive. See for example Index Funds: The 12-Step program For Active Investors, by Mark T. Hebner, (IFA Publishing, (c) 2007). The data cited in the Hebner book shows that the large blend vs. small value performance over 1926-2006 is 10.65% vs. 14.50%, an annual performance differential of 3.85%. This expected level of outperformance is an extremely attractive prospect for the long-term investor. However, practical implementation of a true minimum-variance portfolio may be very difficult.
The variance of a portfolio can be computed from the covariance matrix and portfolio weights as shown in, for example Investment Analysis and Portfolio Management, 4th. ed. by Jerome B. Cohen, Edward D. Zinbarg, and Arthur Zeikel, (Richard D. Irwin Inc., (c) 1982) Chapter 4, Appendix B.
Seeking an unconstrained minimum variance solution may be unstable (i.e. may lead to large positive and negative weights) and require more data than can be easily assembled. For example, portfolio optimization (as described in e.g. equation 2.6 of Best) requires inversion of the asset return covariance matrix. However, it is well-known that an n-by-n covariance matrix based on fewer than n observations will be rank-deficient and hence not invertible, and that even with n or more observations the matrix may be singular or nearly singular.
Take for example the problem of finding a minimum variance portfolio for the Russell 2000 Index, which is based on performance data for approximately 2,000 small-cap stocks. If daily closing data is assembled for each stock, then under the assumption of 250 trading days per year, it will take at least eight years to assemble a full-rank covariance matrix. This period of time is longer than a typical business cycle, raising the possibility that by the time the data is collected, it may be too late to make timely investment decisions based on it.
Additionally, for all but the largest funds, maintaining a position in each of 2,000 stocks may be expensive, and so some kind of sampling approach (investing in only a subset of the 2,000 stocks) is necessary. A method based on picking a relatively small number of stocks for the portfolio and adding them incrementally to the portfolio could substantially reduce transaction costs.
An equal-weighting approach reduces the difficulty of the optimization problem, avoids unstable and wildly differing portfolio weights for stocks in the portfolio, and automatically constrains the portfolio to be long-only. It can also be justified on the basis that more sophisticated methods may give statistically-indistinguishable results unless measurement is over a very long timescale. See Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?, by Victor DeMiguel, Lorenzo Garlappi, and Raman Uppal, (c) 2007 Oxford University Press, available at: http://faculty.london.edu/avmiguel/DeMiguel-Garlappi-Uppal-RFS.pdf
A refinement of the 1/N approach is to start with equal weights and then modify them within bounds (say 50% to 150% of the equal weights), to further reduce volatility. This “modified equal-weighting” approach avoids the unstable portfolio weights arising from an unconstrained minimization approach while making fuller use of the data, and is computationally within a constant factor of the amount of time it takes to determine the volatility of the 1/N portfolio.
But even after applying the equal-weighting approach or the modified equal-weighting approach, the solution space is very large. If we are selecting 50 equally-weighted stocks from a set of 500 (for example) with the objective of minimizing variance, there are C(500,50)=2.3×1069 combinations to consider, while if selecting 50 stocks from a set of 2000 there are C(2000,50)=2.0×10100. Clearly, exhaustive search of all possible combinations is impractical.
However, if there were a practical, even if approximate, solution to this problem, it could be iterated: pick the next-lowest variance basket of 50 stocks repeatedly until a stopping criterion (such as selection of 25% of all stocks in the original set) is satisfied.
Accordingly there is a long-felt need for a system allowing incremental construction of an approximate minimum-variance modified-equal-weight portfolio of size m, given an n by n possibly rank-deficient covariance matrix of asset returns, where m is less than n.
It will be appreciated that various of the above-disclosed and other features and functions, or alternatives thereof, may be desirably combined into many other different systems or applications. Various presently unforeseen or unanticipated alternatives, modifications, variations, or improvements therein may be subsequently made by those skilled in the art which are also intended to be encompassed by the following claims.